Limits, Continuity, Differentiability & Mean Value Theorems

FUNCTIONS 

Definition:

We can define a function as a special relation which maps each element of set A with one and only one element of set B. Both the sets A and B must be nonempty. A function defines a particular output for a particular input.

Hence, f: A → B is a function such that for a ∈ A there is a unique element b ∈ B such that (a, b) ∈ f

Let f : N → N such that f(N) = 2x+1

Then domain = N,

Range = Odd natural no. = {1, 3, 5, 7……},

Co-domain = N

LIMIT OF A FUNCTION:

Let us consider a function f(x) defined in an interval l.If we see the behaviour of f(x) become closer and closer to a number l as x → a then l is said to be limit of f(x)at x=a.

Left-Hand Limit –

Let  function f(x) is said to approach l as x → a from left if for an arbitrary positive small number ε ,a small positive number δ (depends on ε) such that

Right-Hand Limit –

Let function f(x) is said to approach l as x → a from right if for an arbitrary positive small number ε, a small positive number δ (depends on ε) such that

Important Results on Limits:

INDETERMINATE FORMS:

Let us consider a function:

  L- Hospital Rule:

 CONTINUITY:

A function y = f(x) is said to be continuous if the graph of the function is a continuous curve. On the other hand, if a curve is broken at some point say x = a, we say that the function is not continuous or discontinuous.

Definition:

A function f(x) is said to be continuous at x = a, if and only if the following three conditions are satisfied:

Properties of continuous functions:

(i) A function which is continuous in a closed interval is also bounded in that interval.

(ii) A continuous function which has opposite signs at two points vanishes atleast once between these points and vanishing point is called root of the function.

(iii) A continuous function f(x) in the closed interval [a, b] assumes at least once every value between f(a) and f(b),

     it being assumed that  f(a) ≠ f(b).

DIFFERENTIABILITY:

Chain Rule of differentiability:

Let f and g be functions defined on an interval l and f, g are differentiable at

x = a ϵ l then

(i) F ± G is differentiable and (F ± G)’ (a) = F’(a) ± G’(a).

(ii) cF is differentiable and (cF)’ (a) = c F’ (a) : c ϵ R.

(iii) F.G is differentiable and (FG)’(a) = F’(a)G(a) + F(a) G’(a).

Mean Value Theorems:

Rolle’s Theorem:

If

(i)  f(x) is continuous is the closed interval [a, b],

(ii)  f’(x) exists for every value of x in the open interval (a, b) and

(iii)  f (a) = f(b), then there is at least one value c of x in (a, b) such that f’ (c) = 0.

Lagrange’s Mean-Value Theorem:

If

(i) f(x) is continuous in the closed interval [a, b], and

(ii) f’(x) exists in the open interval (a, b),

 then there is at least there is at one value c of x (a, b),

such that.

Cauchy’s Mean-value theorem:

If (i) f(x) and g(x) be continuous in [a, b]

(ii) f’(x) and g’(x) exist in (a, b) and

(iii) g’(x) ≠ 0 for any value of x in (a, b),

Then there is at least one value c of x in (a, b),

such that,